20777_Problem_solving_Year_6_Paterns_and_algebra_Fractions_and_decimals_Using_units_of_measurement

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YEAR 6 

PROBLEM-SOLVING 

IN MATHEMATICS 

Patterns and algebra/ 

Fractions and decimals/ 

Using units of measurement 

Two-time winner of the Australian 

Primary Publisher of the Year Award

Problem-solving in mathematics 

(Book G) 

Published by R.I.C. Publications ® 2008 

Copyright © George Booker and 

Denise Bond 2007 

RIC–20777 

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FOREWORD 

Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers 

of students from the early years to the end of middle school and into secondary school. The series of problems, 

discussions of ways to understand what is being asked and means of obtaining solutions have been built up to 

improve the problem-solving performance and persistence of all students. It is a fundamental belief of the authors 

that it is critical that students and teachers engage with a few complex problems over an extended period rather than 

spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow students 

time to review and discuss what is required in the problem-solving process before moving to another and different 

problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in 

implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and 

selected over many years’ experience with students at all levels of mathematical talent and persistence, as well as 

in discussions with teachers in classrooms, professional learning and university settings. 

Problem-solving does not come easily to most people, 

so learners need many experiences engaging with 

problems if they are to develop this crucial ability. As 

they grapple with problem, meaning and find solutions, 

students will learn a great deal about mathematics 

and mathematical reasoning; for instance, how to 

organise information to uncover meanings and allow 

connections among the various facets of a problem 

to become more apparent, leading to a focus on 

organising what needs to be done rather than simply 

looking to apply one or more strategies. In turn, this 

extended thinking will help students make informed 

choices about events that impact on their lives and to 

interpret and respond to the decisions made by others 

at school, in everyday life and in further study. 

Student and teacher pages 

The student pages present problems chosen with a 

particular problem-solving focus and draw on a range 

of mathematical understandings and processes. 

For each set of related problems, teacher notes and 

discussion are provided, as well as indications of 

how particular problems can be examined and solved. 

Answers to the more straightforward problems and 

detailed solutions to the more complex problems 

ensure appropriate explanations, the use of the 

pages, foster discussion among students and suggest 

ways in which problems can be extended. Related 

problems occur on one or more pages that extend the 

problem’s ideas, the solution processes and students’ 

understanding of the range of ways to come to terms 

with what problems are asking. 

At the top of each teacher page, there is a statement 

that highlights the particular thinking that the 

problems will demand, together with an indication 

of the mathematics that might be needed and a list 

of materials that could be used in seeking a solution. 

A particular focus for the page or set of three pages 

of problems then expands on these aspects. Each 

book is organised so that when a problem requires 

complicated strategic thinking, two or three problems 

occur on one page (supported by a teacher page with 

detailed discussion) to encourage students to find 

a solution together with a range of means that can 

be followed. More often, problems are grouped as a 

series of three interrelated pages where the level of 

complexity gradually increases, while the associated 

teacher page examines one or two of the problems in 

depth and highlights how the other problems might be 

solved in a similar manner. 

R.I.C. Publications ® www.ricpublications.com.au Problem-solving in mathematics 

iii

FOREWORD 

Each teacher page concludes with two further aspects 

critical to successful teaching of problem-solving. A 

section on likely difficulties points to reasoning and 

content inadequacies that experience has shown may 

well impede students’ success. In this way, teachers 

can be on the look out for difficulties and be prepared 

to guide students past these potential pitfalls. The 

final section suggests extensions to the problems to 

enable teachers to provide several related experiences 

with problems of these kinds in order to build a rich 

array of experiences with particular solution methods; 

for example, the numbers, shapes or measurements 

in the original problems might change but leave the 

means to a solution essentially the same, or the 

context may change while the numbers, shapes or 

measurements remain the same. Then numbers, 

shapes or measurements and the context could be 

changed to see how the students handle situations 

that appear different but are essentially the same 

as those already met and solved. Other suggestions 

ask students to make and pose their own problems, 

investigate and present background to the problems 

or topics to the class, or consider solutions at a more 

general level (possibly involving verbal descriptions 

and eventually pictorial or symbolic arguments). 

In this way, not only are students’ ways of thinking 

extended but the problems written on one page are 

used to produce several more problems that utilise 

the same approach. 

Mathematics and language 

The difficulty of the mathematics gradually increases 

over the series, largely in line with what is taught 

at the various year levels, although problem-solving 

both challenges at the point of the mathematics 

that is being learned as well as provides insights 

and motivation for what might be learned next. For 

example, the computation required gradually builds 

from additive thinking, using addition and subtraction 

separately and together, to multiplicative thinking, 

where multiplication and division are connected 

conceptions. More complex interactions of these 

operations build up over the series as the operations 

are used to both come to terms with problems’ 

meanings and to achieve solutions. Similarly, twodimensional 

geometry is used at first but extended 

to more complex uses over the range of problems, 

then joined by interaction with three-dimensional 

ideas. Measurement, including chance and data, also 

extends over the series from length to perimeter, and 

from area to surface area and volume, drawing on 

the relationships among these concepts to organise 

solutions as well as giving an understanding of the 

metric system. Time concepts range from interpreting 

timetables using 12-hour and 24-hour clocks while 

investigations related to mass rely on both the concept 

itself and practical measurements. 

The language in which the problems are expressed is 

relatively straightforward, although this too increases 

in complexity and length of expression across the books 

in terms of both the context in which the problems 

are set and the mathematical content that is required. 

It will always be a challenge for some students 

to ‘unpack’ the meaning from a worded problem, 

particularly as problems’ context, information and 

meanings expand. This ability is fundamental to the 

nature of mathematical problem-solving and needs to 

be built up with time and experiences rather than be 

iv 

Problem-solving in mathematics www.ricpublications.com.au R.I.C. Publications ®

FOREWORD 

diminished or left out of the problems’ situations. One 

reason for the suggestion that students work in groups 

is to allow them to share and assist each other with 

the tasks of discerning meanings and ways to tackle 

the ideas in complex problems through discussion, 

rather than simply leaping into the first ideas that 

come to mind (leaving the full extent of the problem 

unrealised). 

An approach to solving problems 

Try 

an approach 

Explore 

means to a solution 

Analyse 

the problem 

The careful, gradual development of an ability to 

analyse problems for meaning, organising information 

to make it meaningful and to make the connections 

among them more meaningful in order to suggest 

a way forward to a solution is fundamental to the 

approach taken with this series, from the first book 

to the last. At first, materials are used explicitly to 

aid these meanings and connections; however, in 

time they give way to diagrams, tables and symbols 

as understanding and experience of solving complex, 

engaging problems increases. As the problem forms 

expand, the range of methods to solve problems 

is carefully extended, not only to allow students to 

successfully solve the many types of problems, but 

also to give them a repertoire of solution processes 

that they can consider and draw on when new 

situations are encountered. In turn, this allows them 

to explore one or other of these approaches to see 

whether each might furnish a likely result. In this way, 

when they try a particular method to solve a new 

problem, experience and analysis of the particular 

situation assists them to develop a full solution. 

Not only is this model for the problem-solving process 

helpful in solving problems, it also provides a basis for 

students to discuss their progress and solutions and 

determine whether or not they have fully answered 

a question. At the same time, it guides teacher 

questions of students and provides a means of seeing 

underlying mathematical difficulties and ways in 

which problems can be adapted to suit particular 

needs and extensions. Above all, it provides a common 

framework for discussions between a teacher and 

group or whole class to focus on the problem-solving 

process rather than simply on the solution of particular 

problems. Indeed, as Alan Schoenfeld, in Steen L (Ed) 

Mathematics and democracy (2001), states so well, in 

problem-solving: 

getting the answer is only the beginning rather than 

the end … an ability to communicate thinking is 

equally important. 

We wish all teachers and students who use these 

books success in fostering engagement with problemsolving 

and building a greater capacity to come to 

terms with and solve mathematical problems at all 

levels. 

George Booker and Denise Bond 


v

CONTENTS 

Foreword .................................................................. iii – v 

Contents .......................................................................... vi 

Introduction ........................................................... vii – xix 

A note on calculator use ................................................ xx 

Teacher notes................................................................... 2 

Surface area...................................................................... 3 

Volume and surface area................................................. 4 

Surface area and volume................................................. 5 

Teacher notes................................................................... 6 

The farmers market.......................................................... 7 

Teacher notes................................................................... 8 

Profit and loss.................................................................. 9 

Calculator patterns........................................................ 10 

Puzzle scrolls.................................................................. 11 

Teacher notes................................................................. 12 

Weather or not............................................................... 13 

Showtime....................................................................... 14 

Probably true.................................................................. 15 

Teacher notes................................................................. 16 

Wilderness explorer....................................................... 17 

Teacher notes................................................................. 18 

Changing lockers............................................................ 19 

Cycle days...................................................................... 20 


Teacher notes................................................................. 22 

Office hours.................................................................... 23 

At the office................................................................... 24 

Out of office................................................................... 25 

Teacher notes................................................................. 26 

Number patterns............................................................ 27 

Teacher notes................................................................. 28 

Magic squares............................................................... 29 

Sudoku........................................................................... 30 

Alphametic puzzles........................................................ 31 

Teacher notes................................................................. 32 

At the shops................................................................... 33 

The plant nursery........................................................... 34 

On the farm.................................................................... 35 

Teacher notes................................................................. 36 

Fisherman’s wharf.......................................................... 37 

Teacher notes................................................................. 38 

Making designs.............................................................. 39 

Squares and rectangles................................................. 40 

Designer squares........................................................... 41 

Teacher notes................................................................. 42 

How many?..................................................................... 43 

How far?......................................................................... 44 

How much?..................................................................... 45 

Teacher notes................................................................. 46 

Prospect Plains............................................................... 47 

Teacher notes................................................................. 48 

Money matters............................................................... 49 

Scoring points................................................................ 50 


Teacher notes.................................................................. 52 

Riding to work................................................................. 53 

Bike tracks....................................................................... 54 

Puzzle scrolls................................................................... 55 

Teacher notes.................................................................. 56 

Farm work....................................................................... 57 

Farm produce.................................................................. 58 

Selling fish...................................................................... 59 

Teacher notes.................................................................. 60 

Rolling along................................................................... 61 

Solutions .................................................................62–71 

Isometric resource page .............................................. 72 

0–99 board resource page ........................................... 73 

4-digit number expander resource page (x 3) .............. 74 

10 mm x 10 mm grid resource page ........................... 75 

15 mm x 15 mm grid resource page ............................ 76 

Triangular grid resource page ...................................... 77 

vi 


TEACHER NOTES 

Problem-solving 

To use strategic thinking to solve problems 

Materials 

grid paper, counters in several different colours 

Focus 

These pages explore more complex problems in which the 

most difficult step is to find a way of coming to terms with 

what the problem is asking. Using a table or diagram to 

explore the situation will assist in seeing all the conditions 

that need to be considered. 

Discussion 

Page 19 

The first problem can be solved using counters on a 

grid or colouring the squares to see what is happening. 

1 2 3 4 5 6 7 8 9 

The number of possibilities can then be seen directly or 

patterns can be sought. 

Analysis of the patterns shows that the squares represent 

the factors of the number of the column in which they 

occur. Determining the factors of each number 1–50 shows 

that 48 has the most factors or entries in a column. The 

other questions are solved by considering prime numbers, 

squares of prime numbers and systematically examining 

the pairs of factors in each number. A discussion of prime 

numbers is on page 64. 

Page 20 

For the first problem, consider the distances covered each 

15 minutes (a table or list would help). After 15 minutes, 

Jane would walk 1 km and have 15 km left, while Jenny 

would ride 3 km and have 13 km left. After 30 minutes, 


would ride 6 km and have 10 km left. After 45 minutes, 


would ride 9 km and have only 7 km left. Jenny would 

then be home first. Jenny should leave the bicycle after 30 

minutes and they would both arrive home together after 

150 minutes. (Tables to illustrate this solution are on page 

64). 

For the second problem, organising the information on to a 

table will supply a solution. The slower speed requires an 

additional 10 km, the faster speed covers 15 km too much: 

hours 

distance @ 

10 km/h 

distance 

to park 

distance @ 

15 km/h 

distance 

to park 

1 10 20 15 0 

2 20 30 30 15 

3 30 40 45 30 

4 40 50 60 45 

5 50 60 75 60 

After 5 hours, the distance is 60 km so 12 km/h would 

arrive at the exact time. The third problem is done the 

same way – the required speed is 9.6 km/h for 5 hours. 

Page 21 

The puzzle scrolls contain a number of different problems 

all involving strategic thinking to find possible solutions. In 

most cases students will find tables, lists and diagrams are 

needed to manage the data while exploring the different 

possibilities. For example, the investigation about bread 

rolls could involve students list the multiples of 60 in a 

table with the total to make $11 and seeing which is a 

multiple of 85. 

Multiple 

of $0.60 

Possible difficulties 

• Not using a diagram or table to come to terms with 

the problem conditions 

• Unable to see how to connect the time cycled to the 

distance travelled 

• Considering only some aspects of the puzzle scrolls 

Extension 

• Change the numbers and the scenarios to write other 

problems based on the puzzle scrolls. 

• Use different speeds and times for the problems 

on p. 20. 

Look for a 

multiple of $0.85 

Total 

$.060 $10.40 $11.00 

$1.20 $9.80 $11.00 

18 


CHANGING LOCKERS 

The local college has exactly 1000 students, each of whom has a locker. The lockers 

are along the passageways and numbered 1–1000. To raise money for local charities, 

the students organised a competition with an entry fee of $1.00. 

All 1000 students had to run past, opening or shutting locker doors: 

• the first student opened the door of every locker 

• the second student closed every locker door with an even number 

• the third student changed every third locker, closing those that were open and 

opening those that were closed 

• the fourth student changed every fourth locker, and so on. 

1. The student or students who could predict ahead of time which lockers would be open 

would nominate the charity that would receive the $1000.00. 

(a) What would you predict? 

(e) Use your pattern to work out which 

lockers would be open after all 

1000 students had run past. 

(b) Would any lockers remain open 

after 10 students had passed along 

the rows of lockers? 

(f) Can you find some lockers that 

only changed twice? 

(c) Which lockers would be open 

after 50 students had changed 50 

lockers? 

(g) Can you see a pattern for the 

numbers on these lockers? 

(d) Can you see a pattern for the 

numbers of the open lockers? 

(h) What is the largest number on a 

locker that changed only twice? 


19

CYCLE DAYS 

Jane and Jenny rode their bicycles to the bay for a picnic. On the way back, when they 

were 16 km from home, the front wheel on Jane’s bicycle was damaged in a large 

pothole and could no longer be ridden. In order for them both to get home in good 

time, they decided to share the walking and bike riding. 

1. At first, Jane would walk while Jenny would ride her bicycle. After some time, Jenny 

would leave her bicycle at the side of the road and continue on foot. When Jane 

reached the bicycle, she would then ride it home. Jane walks at 4 km per hour and 

cycles at 10 km per hour, while Jenny walks at 5 km per hour and cycles at 12 km per 

hour. How long should Jenny ride her bicycle for both of them to arrive home at the 

same time? 

2. When Jane’s bicycle was repaired, they decided to go for another bike ride. This time, 

they would set off separately and meet at the Jetty at midday, then set up their picnic 

in the park next door. Jenny knew that if she took it easy and rode at 10 km per hour, 

she would be an hour late, but if she really pushed herself and rode at 15 km per hour, 

she would arrive an hour early. At what speed should she ride in order to arrive just on 

time? 

3. Jane did not have quite as far to travel so she knew she did not have to ride as hard as 

Jenny. If she rode at 12 km per hour, she would arrive an hour early, but if she rode at 

8 km per hour, she would be an hour late. At what speed should Jane ride to get there 

at the same time as Jenny? 

4. At what time did each girl leave her home? 

20 


PUZZLE SCROLLS 

1. $65 000 is divided among 5 

brothers with each one getting 

$3000 more than their younger 

sibling. How much will the 

youngest brother get? 

2. A rectangular table is twice as 

long as it is wide. If it was 3 m 

shorter and 3 m wider it would 

be a square. What size is the 

table? 

3. How many blocks would you 

need to make this stack of cubes 

into a cube 6 blocks high? 

4. I spent $11.00 at the bakery 

buying bread rolls for 60c and 

85c. How many of each roll did I 

buy? 

5. How often in a 12-hour period 

does the sum of the digits on a 

digital clock equal 9? 

6. A triangle has a perimeter of 80 

cm. Two of its sides are equal 

and the third side is 8 cm more 

than the equal sides. 

03:24 

ON 

OFF 

GIGA-BLASTER 

What is the length of the third 

side? 


21

SOLUTIONS 

Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and students to solve 

problems in this way. 

CHANGING LOCKERS................................................ page 19 

1. (a) Answers will vary. 

Colour squares or place counters on a grid to show 

when a door is open: 

A door is changed an odd number of times to 

remain open. These locker numbers are square 

number (one number as a factor twice, so an odd 

number of factors): 1, 4, 9, 16, 25, 49, ... 

A prime number has only two factors and these 

lockers will always be closed. 

(b) 1, 4, 9 

(c) 1, 4, 9, 16, 25, 49 

(d) Square numbers have an odd number of factors and 

these lockers will be open. 

(e) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 

225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 

625, 676, 729, 784, 841, 900, 961 

(f) 2, 3, 5, 7, 11, 13 

(g) 

(h) 

1 2 3 4 5 6 7 8 9 

They are prime numbers and have only 2 factors 

The Sieve of Eratosthenes shows that after 2 and 

3, all primes must be 1 more or less than a multiple 

of 6 (although some of these numbers are not 

prime – this was discussed in Book F). 

1 2 3 4 5 6 7 

8 9 10 11 12 13 

14 15 16 17 18 19 

20 21 22 23 24 25 

26 27 28 29 30 31 

32 33 34 

=166 x 6 = 996 is the closest multiple of 6 to 1000, 

so 997 is the largest prime. 

997 is the largest locker number that is only 

changed twice. 

CYCLE DAYS............................................................... page 20 

1. Looking at their rates of walking and cycling, in 15 

minutes Jane walks 1 km and Jenny cycles 3 km: 

Jane walking 

minutes 

Jenny cycling 

distance 

walked 

distance left to 

cycle 

They can arrive home together after 2 hours 30 minutes. 

Jenny cycles for 30 minutes and walks the remaining 10 

km in 2 hours. Jane walks for 90 minutes to reach the 

bicycle and cycles the remaining 10 km in 1 hour. 

Jenny should ride her bicycle for 30 minutes 

time taken 

15 1 15 1 hour 45 minutes 


45 3 13 2 hours 3 minutes 




minutes 

distance 

cycled 

distance left to 

walk 

time taken 





Jane 90 minutes walking 1 hour cycling 

km 0 1 2 3 4 5 6 7 8 9 10 11 12 13 15 15 16 

Jenny 30 minutes cycling 2 hours walking 

2. The slower speed requires an additional 10 km, the 

faster speed covers 15 km too much: 

hours 

distance @ 

10km/her 

distance to 

park 

distance @ 

15 km/her 

distance 

to park 

1 10 20 15 0 

2 20 30 30 15 

3 30 40 45 30 

4 40 50 60 45 

5 50 60 75 60 

After 5 hours, the distance to the park at each speed is 

the same – 60 km. 

She would arrive at the exact time cycling at 

12 km per hour. 

64 


SOLUTIONS 

Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and students to solve 

problems in this way. 

3. The slower speed requires an additional 8 km, the faster 

speed cover 12 km too much: 

hours 

distance @ 

10km/her 

distance to 

park 

distance @ 

15 km/her 

distance 

to park 

1 10 20 15 0 

2 20 30 30 15 

3 30 40 45 30 

4 40 50 60 45 

5 50 60 75 60 

After 5 hours, the distance to the park at each speed is 

the same – 48 km. 

She would arrive at the exact time cycling at 9.6 km per 

hour 

4. Both Jenny and Jane should leave at 7:00 am 

PUZZLE SCROLLS ...................................................... page 21 

1. $7000 

2. width 6 m, length 12 m 

3. 210 blocks 

4. 7 @ 60 c, 8 @ 85 c 

5. 57 times. Student answers will vary and it is unlikely 

they will find them all. 

6. 32 cm 


65

10 mm x 10 mm GRID RESOURCE PAGE 


75

15 mm x 15 mm GRID RESOURCE PAGE 

76

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